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The initial-value problem for the forced Korteweg-de Vries equation*

Published online by Cambridge University Press:  14 November 2011

Jerry L. Bona  and
Bing-Yu Zhang
Jerry L. Bona
Affiliation:
Department of Mathematics and Computational & Applied Mathematics Program, University of Texas, Austin, TX 78712, U.S.A.
Bing-Yu Zhang
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, U.S.A.
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Extract

The initial-value problem for the Korteweg-de Vries equation with a forcing term has recently gained prominence as a model for a number of interesting physical situations. At the same time, the modern theory for the initial-value problem for the unforced Korteweg-de Vries equation has taken great strides forward. The mathematical theory pertaining to the forced equation is currently set in narrow function classes and has not kept up with recent advances for the homogeneous equation. This aspect is rectified here with the development of a theory for the initial-value problem for the forced Korteweg-de Vries equation that entails weak assumptions on both the initial wave configuration and the forcing. The results obtained include analytic dependence of solutions on the auxiliary data and allow the external forcing to lie in function classes sufficiently large that a Dirac δ-function or its derivative is included. Analyticity is proved by an infinite-dimensional analogue of Picard iteration. A consequence is that solutions may be approximated arbitrarily well on any bounded time interval by solving a finite number of linear initial-value problems.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 3 , 1996 , pp. 571 - 598
Copyright
Copyright © Royal Society of Edinburgh 1996

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